To demonstrate set union and intersection to her class, Mrs. Putnam asked for three students to each write down a set of numbers.

After they had done so, she looked at their sets and told the class, "the union of these three sets is the first ten counting numbers, but their intersection is empty!"

How many triples (A, B, C) of sets are there such that

A U B U C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
and
A ∩ B ∩ C = {} ?

You are right again, SilverKnight. (I am beginning to feel like one of the hapless students of Professor Kingsfield in "The Paper Chase")

(A int B) int C = {}

If C has just 1 number (there are 10 such sets), then any two sets A and B, neither of which have that number, will work. Each of the other nine numbers can be in A, in B, or both
10*(3^9)

If C has 2 numbers (10*90 sets),
90*(3^8)

If C has 3 numbers (10*9*8=720)
720*(3^7)

If C has 4 numbers, (10*9*8*7=5040)
5040*(3^6)

If C has 5 numbers (10*9*8*7*6 = 30240)
30240*(3^5)

If C has 6 numbers (10*9*8*7*6*5 = 151200)
151200*(3^4)

If C has 7 numbers (10*9*8*7*6*5*4 = 604800)
604800*(3^3)

If C has 8 numbers (10*9*8*7*6*5*4*3=1814400)
1814400*(3^2)



Edited on February 10, 2004, 11:41 am

Posted by Penny on 2004-02-10 11:31:50